TSTP Solution File: ALG177+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ALG177+1 : TPTP v8.1.2. Released v2.7.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n028.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 16:42:24 EDT 2023
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ALG177+1 : TPTP v8.1.2. Released v2.7.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.34 % Computer : n028.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Mon Aug 28 05:50:24 EDT 2023
% 0.17/0.34 % CPUTime :
% 0.20/0.40 Command-line arguments: --no-flatten-goal
% 0.20/0.40
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 Take the following subset of the input axioms:
% 0.20/0.41 fof(ax3, axiom, ?[U]: (sorti1(U) & ![V]: (sorti1(V) => op1(U, V)!=V))).
% 0.20/0.41 fof(ax4, axiom, ~?[U2]: (sorti2(U2) & ![V2]: (sorti2(V2) => op2(U2, V2)!=V2))).
% 0.20/0.41 fof(co1, conjecture, (![U2]: (sorti1(U2) => sorti2(h(U2))) & ![V2]: (sorti2(V2) => sorti1(j(V2)))) => ~(![W]: (sorti1(W) => ![X]: (sorti1(X) => h(op1(W, X))=op2(h(W), h(X)))) & (![Y]: (sorti2(Y) => ![Z]: (sorti2(Z) => j(op2(Y, Z))=op1(j(Y), j(Z)))) & (![X1]: (sorti2(X1) => h(j(X1))=X1) & ![X2]: (sorti1(X2) => j(h(X2))=X2))))).
% 0.20/0.41
% 0.20/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.41 fresh(y, y, x1...xn) = u
% 0.20/0.41 C => fresh(s, t, x1...xn) = v
% 0.20/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.41 variables of u and v.
% 0.20/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.41 input problem has no model of domain size 1).
% 0.20/0.41
% 0.20/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.41
% 0.20/0.41 Axiom 1 (ax3): sorti1(u) = true2.
% 0.20/0.41 Axiom 2 (ax4): fresh10(X, X, Y) = v(Y).
% 0.20/0.41 Axiom 3 (ax4_1): fresh9(X, X, Y) = true2.
% 0.20/0.41 Axiom 4 (co1): fresh8(X, X, Y) = true2.
% 0.20/0.41 Axiom 5 (co1_3): fresh5(X, X, Y) = true2.
% 0.20/0.41 Axiom 6 (co1_2): fresh2(X, X, Y) = Y.
% 0.20/0.41 Axiom 7 (ax4): fresh10(sorti2(X), true2, X) = op2(X, v(X)).
% 0.20/0.41 Axiom 8 (ax4_1): fresh9(sorti2(X), true2, X) = sorti2(v(X)).
% 0.20/0.41 Axiom 9 (co1): fresh8(sorti1(X), true2, X) = sorti2(h(X)).
% 0.20/0.41 Axiom 10 (co1_3): fresh5(sorti2(X), true2, X) = sorti1(j(X)).
% 0.20/0.41 Axiom 11 (co1_4): fresh3(X, X, Y, Z) = j(op2(Y, Z)).
% 0.20/0.41 Axiom 12 (co1_2): fresh2(sorti1(X), true2, X) = j(h(X)).
% 0.20/0.41 Axiom 13 (co1_4): fresh4(X, X, Y, Z) = op1(j(Y), j(Z)).
% 0.20/0.41 Axiom 14 (co1_4): fresh4(sorti2(X), true2, Y, X) = fresh3(sorti2(Y), true2, Y, X).
% 0.20/0.41
% 0.20/0.41 Lemma 15: sorti2(h(u)) = true2.
% 0.20/0.41 Proof:
% 0.20/0.41 sorti2(h(u))
% 0.20/0.41 = { by axiom 9 (co1) R->L }
% 0.20/0.41 fresh8(sorti1(u), true2, u)
% 0.20/0.41 = { by axiom 1 (ax3) }
% 0.20/0.41 fresh8(true2, true2, u)
% 0.20/0.41 = { by axiom 4 (co1) }
% 0.20/0.41 true2
% 0.20/0.41
% 0.20/0.41 Lemma 16: sorti2(v(h(u))) = true2.
% 0.20/0.41 Proof:
% 0.20/0.41 sorti2(v(h(u)))
% 0.20/0.41 = { by axiom 8 (ax4_1) R->L }
% 0.20/0.41 fresh9(sorti2(h(u)), true2, h(u))
% 0.20/0.41 = { by lemma 15 }
% 0.20/0.41 fresh9(true2, true2, h(u))
% 0.20/0.41 = { by axiom 3 (ax4_1) }
% 0.20/0.41 true2
% 0.20/0.41
% 0.20/0.41 Goal 1 (ax3_1): tuple(op1(u, X), sorti1(X)) = tuple(X, true2).
% 0.20/0.41 The goal is true when:
% 0.20/0.41 X = j(v(h(u)))
% 0.20/0.41
% 0.20/0.41 Proof:
% 0.20/0.41 tuple(op1(u, j(v(h(u)))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 6 (co1_2) R->L }
% 0.20/0.41 tuple(op1(fresh2(true2, true2, u), j(v(h(u)))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 1 (ax3) R->L }
% 0.20/0.41 tuple(op1(fresh2(sorti1(u), true2, u), j(v(h(u)))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 12 (co1_2) }
% 0.20/0.41 tuple(op1(j(h(u)), j(v(h(u)))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 13 (co1_4) R->L }
% 0.20/0.41 tuple(fresh4(true2, true2, h(u), v(h(u))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by lemma 16 R->L }
% 0.20/0.41 tuple(fresh4(sorti2(v(h(u))), true2, h(u), v(h(u))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 14 (co1_4) }
% 0.20/0.41 tuple(fresh3(sorti2(h(u)), true2, h(u), v(h(u))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by lemma 15 }
% 0.20/0.41 tuple(fresh3(true2, true2, h(u), v(h(u))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 11 (co1_4) }
% 0.20/0.41 tuple(j(op2(h(u), v(h(u)))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 7 (ax4) R->L }
% 0.20/0.41 tuple(j(fresh10(sorti2(h(u)), true2, h(u))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by lemma 15 }
% 0.20/0.41 tuple(j(fresh10(true2, true2, h(u))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 2 (ax4) }
% 0.20/0.41 tuple(j(v(h(u))), sorti1(j(v(h(u)))))
% 0.20/0.41 = { by axiom 10 (co1_3) R->L }
% 0.20/0.41 tuple(j(v(h(u))), fresh5(sorti2(v(h(u))), true2, v(h(u))))
% 0.20/0.41 = { by lemma 16 }
% 0.20/0.41 tuple(j(v(h(u))), fresh5(true2, true2, v(h(u))))
% 0.20/0.41 = { by axiom 5 (co1_3) }
% 0.20/0.41 tuple(j(v(h(u))), true2)
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41
% 0.20/0.41 RESULT: Theorem (the conjecture is true).
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